↑

Properties

Label 2-161-161.100-c1-0-1
Degree 22
Conductor 161161
Sign −0.366−0.930i-0.366 - 0.930i
Analytic cond. 1.285591.28559
Root an. cond. 1.133831.13383
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.733 − 0.576i)2-s + (−0.358 + 0.503i)3-s + (−0.266 − 1.09i)4-s + (−3.83 + 0.738i)5-s + (0.553 − 0.162i)6-s + (1.47 + 2.19i)7-s + (−1.21 + 2.65i)8-s + (0.856 + 2.47i)9-s + (3.23 + 1.66i)10-s + (−1.69 + 1.32i)11-s + (0.648 + 0.259i)12-s + (0.00277 − 0.00178i)13-s + (0.190 − 2.46i)14-s + (1.00 − 2.19i)15-s + (0.413 − 0.213i)16-s + (−5.42 − 5.17i)17-s + ⋯
L(s)  = 1  + (−0.518 − 0.407i)2-s + (−0.206 + 0.290i)3-s + (−0.133 − 0.548i)4-s + (−1.71 + 0.330i)5-s + (0.225 − 0.0663i)6-s + (0.555 + 0.831i)7-s + (−0.428 + 0.939i)8-s + (0.285 + 0.824i)9-s + (1.02 + 0.527i)10-s + (−0.509 + 0.400i)11-s + (0.187 + 0.0748i)12-s + (0.000769 − 0.000494i)13-s + (0.0508 − 0.657i)14-s + (0.258 − 0.566i)15-s + (0.103 − 0.0532i)16-s + (−1.31 − 1.25i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.366−0.930i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+1/2) L(s)=((−0.366−0.930i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.366−0.930i-0.366 - 0.930i
Analytic conductor: 1.285591.28559
Root analytic conductor: 1.133831.13383
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ161(100,⋅)\chi_{161} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :1/2), −0.366−0.930i)(2,\ 161,\ (\ :1/2),\ -0.366 - 0.930i)

Particular Values

L(1)L(1) ≈\approx 0.173277+0.254431i0.173277 + 0.254431i
L(12)L(\frac12) ≈\approx 0.173277+0.254431i0.173277 + 0.254431i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1+(−1.47−2.19i)T 1 + (-1.47 - 2.19i)T
23 1+(−4.79+0.0906i)T 1 + (-4.79 + 0.0906i)T
good2 1+(0.733+0.576i)T+(0.471+1.94i)T2 1 + (0.733 + 0.576i)T + (0.471 + 1.94i)T^{2}
3 1+(0.358−0.503i)T+(−0.981−2.83i)T2 1 + (0.358 - 0.503i)T + (-0.981 - 2.83i)T^{2}
5 1+(3.83−0.738i)T+(4.64−1.85i)T2 1 + (3.83 - 0.738i)T + (4.64 - 1.85i)T^{2}
11 1+(1.69−1.32i)T+(2.59−10.6i)T2 1 + (1.69 - 1.32i)T + (2.59 - 10.6i)T^{2}
13 1+(−0.00277+0.00178i)T+(5.40−11.8i)T2 1 + (-0.00277 + 0.00178i)T + (5.40 - 11.8i)T^{2}
17 1+(5.42+5.17i)T+(0.808+16.9i)T2 1 + (5.42 + 5.17i)T + (0.808 + 16.9i)T^{2}
19 1+(4.04−3.85i)T+(0.904−18.9i)T2 1 + (4.04 - 3.85i)T + (0.904 - 18.9i)T^{2}
29 1+(4.90−1.44i)T+(24.3−15.6i)T2 1 + (4.90 - 1.44i)T + (24.3 - 15.6i)T^{2}
31 1+(2.62−0.250i)T+(30.4−5.86i)T2 1 + (2.62 - 0.250i)T + (30.4 - 5.86i)T^{2}
37 1+(−0.268−0.775i)T+(−29.0+22.8i)T2 1 + (-0.268 - 0.775i)T + (-29.0 + 22.8i)T^{2}
41 1+(−4.66−5.38i)T+(−5.83+40.5i)T2 1 + (-4.66 - 5.38i)T + (-5.83 + 40.5i)T^{2}
43 1+(2.84+6.23i)T+(−28.1+32.4i)T2 1 + (2.84 + 6.23i)T + (-28.1 + 32.4i)T^{2}
47 1+(−3.03+5.26i)T+(−23.5−40.7i)T2 1 + (-3.03 + 5.26i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.306−6.44i)T+(−52.7−5.03i)T2 1 + (0.306 - 6.44i)T + (-52.7 - 5.03i)T^{2}
59 1+(−9.04−4.66i)T+(34.2+48.0i)T2 1 + (-9.04 - 4.66i)T + (34.2 + 48.0i)T^{2}
61 1+(0.0837+0.117i)T+(−19.9+57.6i)T2 1 + (0.0837 + 0.117i)T + (-19.9 + 57.6i)T^{2}
67 1+(−0.798+0.319i)T+(48.4−46.2i)T2 1 + (-0.798 + 0.319i)T + (48.4 - 46.2i)T^{2}
71 1+(0.271−1.88i)T+(−68.1−20.0i)T2 1 + (0.271 - 1.88i)T + (-68.1 - 20.0i)T^{2}
73 1+(−2.49−10.3i)T+(−64.8+33.4i)T2 1 + (-2.49 - 10.3i)T + (-64.8 + 33.4i)T^{2}
79 1+(0.0546+1.14i)T+(−78.6+7.50i)T2 1 + (0.0546 + 1.14i)T + (-78.6 + 7.50i)T^{2}
83 1+(1.69−1.95i)T+(−11.8−82.1i)T2 1 + (1.69 - 1.95i)T + (-11.8 - 82.1i)T^{2}
89 1+(7.39+0.706i)T+(87.3+16.8i)T2 1 + (7.39 + 0.706i)T + (87.3 + 16.8i)T^{2}
97 1+(1.92+2.22i)T+(−13.8+96.0i)T2 1 + (1.92 + 2.22i)T + (-13.8 + 96.0i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.94869705986565790551552352399, −11.70145790610223529868135615004, −11.16587135851766500197995427105, −10.47453128660293513631044991913, −9.065886262889047908614644283420, −8.196124430712571522079362489155, −7.15273791700332019933832459098, −5.30096507738647818620828404076, −4.39362141745942351491715644754, −2.38648327737454390299589687418, 0.35039510679551985364481709771, 3.68896567736420990869254215249, 4.43232667440606070561524626951, 6.67136658854521105781803568530, 7.44489492464727499995366641636, 8.294033588109313716220534739465, 9.037299714390330856646096414272, 10.87927223491952469937680909421, 11.45267953907139225602925180312, 12.77059206983852914085892790169

Graph of the ZZ-function along the critical line